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Regular polygons, line operators, and elliptic modular surfaces as realization spaces of matroids

Lukas Kühne, Xavier Roulleau

TL;DR

The paper links matroid realization spaces of line arrangements derived from regular polygons to elliptic modular surfaces $Ξ_{1}(n)$ over $X_{1}(n)$, providing a birational bridge and a rational model over $\mathbb{Q}$. It develops a unified framework of line/point operators $\Lambda_{\mathfrak{n},\mathfrak{m}}$ and $\Psi_{\mathfrak{n},\mathfrak{m}}$ that generate and relate realizations, and shows that generic realizations of the associated matroids sit on unique cubic curves. A key finding is that a natural nine-to-one map from $Ξ_{1}(n)$ to the matroid realization space $\mathcal{R}_{n}$ renders $\mathcal{R}_{n}$ birational to $Ξ_{1}(n)$, enabling an efficient geometric interpretation of multiplication by $-2$ on elliptic curves via these realizations. The work further extends to singular cubic cases, periodic line arrangements, and explicit analyses for pentagon and hexagon configurations, yielding new connections to modular surfaces $Ξ_{1}(n)$ and their arithmetic geometry.

Abstract

For an integer $n\geq 7$, we investigate the matroid realization space of a specific deformation of the regular $n$-gon along with its lines of symmetry. It turns out that this particular realization space is birational to the elliptic modular surface $Ξ_{1}(n)$ over the modular curve $X_{1}(n)$. In this way, we obtain a model of $Ξ_{1}(n)$ defined over the rational numbers. Furthermore, a natural geometric operator acts on these matroid realizations. On the elliptic modular surface, this operator corresponds to the multiplication by $-2$ on the elliptic curves. This provides a new geometric approach to computing multiplication by $-2$ on elliptic curves.

Regular polygons, line operators, and elliptic modular surfaces as realization spaces of matroids

TL;DR

The paper links matroid realization spaces of line arrangements derived from regular polygons to elliptic modular surfaces over , providing a birational bridge and a rational model over . It develops a unified framework of line/point operators and that generate and relate realizations, and shows that generic realizations of the associated matroids sit on unique cubic curves. A key finding is that a natural nine-to-one map from to the matroid realization space renders birational to , enabling an efficient geometric interpretation of multiplication by on elliptic curves via these realizations. The work further extends to singular cubic cases, periodic line arrangements, and explicit analyses for pentagon and hexagon configurations, yielding new connections to modular surfaces and their arithmetic geometry.

Abstract

For an integer , we investigate the matroid realization space of a specific deformation of the regular -gon along with its lines of symmetry. It turns out that this particular realization space is birational to the elliptic modular surface over the modular curve . In this way, we obtain a model of defined over the rational numbers. Furthermore, a natural geometric operator acts on these matroid realizations. On the elliptic modular surface, this operator corresponds to the multiplication by on the elliptic curves. This provides a new geometric approach to computing multiplication by on elliptic curves.
Paper Structure (19 sections, 15 theorems, 37 equations, 4 figures, 1 table)

This paper contains 19 sections, 15 theorems, 37 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries on operators and matroids
  3. The operators $\Lambda_{\mathfrak{n,m}}$ and $\Psi_{\mathfrak{n,m}}$
  4. Matroids
  5. The realization space of a matroid
  6. Elliptic modular surfaces
  7. Matroids from regular polygons
  8. Odd number of sides
  9. Even number of sides
  10. Explicit description of $M_{n}$
  11. A point realization of $M_{n}$ is on a unique cubic curve
  12. Arrangements of translates of a point by torsion points
  13. Realization spaces $\mathcal{R}_{n}$ and the modular surfaces $\Xi_{1}(n)$
  14. Further constructions and results
  15. The nodal cubic curve
  16. ...and 4 more sections

Key Result

Theorem 1

Suppose that $n\geq7$. The realization space $\mathcal{R}_{n}=\mathcal{R}(M_{n})$ is two dimensional and irreducible. If $\mathfrak{\mathcal{C}}_{0}\cup\mathfrak{\mathcal{C}}_{1}$ is a generic realization of $M_{n}$, then $\mathfrak{\mathcal{C}}_{2}=\Lambda(\mathfrak{\mathcal{C}}_{1})$ is an arrange

Figures (4)

  • Figure 3.1: Schematic picture of the labeling for $n=7$ and $r=1$.
  • Figure 3.2: A line arrangement (almost) realizing the matroid $M_{7}$.
  • Figure 5.1: A pentagon arrangement and its image by $\Lambda_{\{2\}}^{0}$ in blue.
  • Figure 6.1: The regular hexagon and the axes of symmetries.

Theorems & Definitions (35)

  • Theorem 1
  • Theorem 2
  • Definition 3
  • Remark 4
  • Definition 5
  • Remark 6
  • Theorem 7
  • Lemma 8
  • proof : Proof of Theorem \ref{['Thm:ExiststUniCubCur']}
  • Theorem 9
  • ...and 25 more